36. Consider again the matrix A in Exercise 35. Give conditions on a, b, c, and d such that A has (a) two distinct real eigenvalues, (b) one real eigenvalue, and (c) no real eigenvalues.
36. Consider again the matrix A in Exercise 35. Give conditions on a, b, c, and d such that A has (a) two distinct real eigenvalues, (b) one real eigenvalue, and (c) no real eigenvalues.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Only do #36
![35. (a) Show that the eigenvalues of the 2 × 2 matrix
a b|
=
c d]
a
A =
are the solutions of the quadratic equation
12 – tr(A)A + det A = 0, where tr(A) is the trace
of A. (See page 162.)
(b) Show that the eigenvalues of the matrix A in
part (a) are
|3D
A = }(a + d ± Vla – d)² + 4bc)
(c) Show that the trace and determinant of the matrix A
in part (a) are given by
ta (A) - + λε and det A = λιλε
where A, and A, are the eigenvalues of A.
36. Consider again the matrix A in Exercise 35. Give
conditions on a, b, c, and d such that A has
(a) two distinct real eigenvalues,
(b) one real eigenvalue, and
(c) no real eigenvalues.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Feb88a591-bb84-4ad5-a9d9-9dd6e5ab413a%2F3fb6c180-bbf3-4b86-98ef-d05ae5becbf1%2Fqm1y11_processed.png&w=3840&q=75)
Transcribed Image Text:35. (a) Show that the eigenvalues of the 2 × 2 matrix
a b|
=
c d]
a
A =
are the solutions of the quadratic equation
12 – tr(A)A + det A = 0, where tr(A) is the trace
of A. (See page 162.)
(b) Show that the eigenvalues of the matrix A in
part (a) are
|3D
A = }(a + d ± Vla – d)² + 4bc)
(c) Show that the trace and determinant of the matrix A
in part (a) are given by
ta (A) - + λε and det A = λιλε
where A, and A, are the eigenvalues of A.
36. Consider again the matrix A in Exercise 35. Give
conditions on a, b, c, and d such that A has
(a) two distinct real eigenvalues,
(b) one real eigenvalue, and
(c) no real eigenvalues.
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